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## G = D5×C22.D4order 320 = 26·5

### Direct product of D5 and C22.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×C22.D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — D5×C22.D4
 Lower central C5 — C2×C10 — D5×C22.D4
 Upper central C1 — C22 — C22.D4

Generators and relations for D5×C22.D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=de-1 >

Subgroups: 1358 in 342 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×10], C22, C22 [×2], C22 [×30], C5, C2×C4, C2×C4 [×4], C2×C4 [×23], D4 [×8], C23 [×2], C23 [×17], D5 [×4], D5 [×3], C10, C10 [×2], C10 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×12], C2×D4, C2×D4 [×7], C24 [×2], Dic5 [×5], C20 [×5], D10 [×8], D10 [×17], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4, C22.D4 [×7], C23×C4, C22×D4, C4×D5 [×14], D20 [×2], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×4], C2×C20 [×2], C5×D4 [×2], C22×D5 [×3], C22×D5 [×4], C22×D5 [×10], C22×C10 [×2], C2×C22.D4, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5, C23.D5 [×2], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×6], C2×C4×D5 [×4], C2×D20, D4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C22×C20, D4×C10, C23×D5 [×2], D5×C22⋊C4, D5×C22⋊C4 [×2], D10.12D4 [×2], C22.D20, D5×C4⋊C4 [×2], D10.13D4 [×2], C23.23D10, C23.18D10, C5×C22.D4, D5×C22×C4, C2×D4×D5, D5×C22.D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C2×C22.D4, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D4 [×2], D5×C22.D4

Smallest permutation representation of D5×C22.D4
On 80 points
Generators in S80
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)
(1 49)(2 50)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 34 19 29)(2 35 20 30)(3 31 16 26)(4 32 17 27)(5 33 18 28)(6 36 11 21)(7 37 12 22)(8 38 13 23)(9 39 14 24)(10 40 15 25)(41 61 56 76)(42 62 57 77)(43 63 58 78)(44 64 59 79)(45 65 60 80)(46 66 51 71)(47 67 52 72)(48 68 53 73)(49 69 54 74)(50 70 55 75)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)

G:=sub<Sym(80)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,56)(42,60)(43,59)(44,58)(45,57)(46,51)(47,55)(48,54)(49,53)(50,52)(61,76)(62,80)(63,79)(64,78)(65,77)(66,71)(67,75)(68,74)(69,73)(70,72), (1,49)(2,50)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,34,19,29)(2,35,20,30)(3,31,16,26)(4,32,17,27)(5,33,18,28)(6,36,11,21)(7,37,12,22)(8,38,13,23)(9,39,14,24)(10,40,15,25)(41,61,56,76)(42,62,57,77)(43,63,58,78)(44,64,59,79)(45,65,60,80)(46,66,51,71)(47,67,52,72)(48,68,53,73)(49,69,54,74)(50,70,55,75), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,56)(42,60)(43,59)(44,58)(45,57)(46,51)(47,55)(48,54)(49,53)(50,52)(61,76)(62,80)(63,79)(64,78)(65,77)(66,71)(67,75)(68,74)(69,73)(70,72), (1,49)(2,50)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,34,19,29)(2,35,20,30)(3,31,16,26)(4,32,17,27)(5,33,18,28)(6,36,11,21)(7,37,12,22)(8,38,13,23)(9,39,14,24)(10,40,15,25)(41,61,56,76)(42,62,57,77)(43,63,58,78)(44,64,59,79)(45,65,60,80)(46,66,51,71)(47,67,52,72)(48,68,53,73)(49,69,54,74)(50,70,55,75), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,18),(2,17),(3,16),(4,20),(5,19),(6,11),(7,15),(8,14),(9,13),(10,12),(21,36),(22,40),(23,39),(24,38),(25,37),(26,31),(27,35),(28,34),(29,33),(30,32),(41,56),(42,60),(43,59),(44,58),(45,57),(46,51),(47,55),(48,54),(49,53),(50,52),(61,76),(62,80),(63,79),(64,78),(65,77),(66,71),(67,75),(68,74),(69,73),(70,72)], [(1,49),(2,50),(3,46),(4,47),(5,48),(6,41),(7,42),(8,43),(9,44),(10,45),(11,56),(12,57),(13,58),(14,59),(15,60),(16,51),(17,52),(18,53),(19,54),(20,55),(21,66),(22,67),(23,68),(24,69),(25,70),(26,61),(27,62),(28,63),(29,64),(30,65),(31,76),(32,77),(33,78),(34,79),(35,80),(36,71),(37,72),(38,73),(39,74),(40,75)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,34,19,29),(2,35,20,30),(3,31,16,26),(4,32,17,27),(5,33,18,28),(6,36,11,21),(7,37,12,22),(8,38,13,23),(9,39,14,24),(10,40,15,25),(41,61,56,76),(42,62,57,77),(43,63,58,78),(44,64,59,79),(45,65,60,80),(46,66,51,71),(47,67,52,72),(48,68,53,73),(49,69,54,74),(50,70,55,75)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)])

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 10A ··· 10F 10G 10H 10I 10J 10K 10L 20A ··· 20H 20I ··· 20N order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 5 5 5 5 10 10 20 2 2 2 2 4 4 4 10 10 10 10 20 20 20 2 2 2 ··· 2 4 4 4 4 8 8 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 D4×D5 D5×C4○D4 kernel D5×C22.D4 D5×C22⋊C4 D10.12D4 C22.D20 D5×C4⋊C4 D10.13D4 C23.23D10 C23.18D10 C5×C22.D4 D5×C22×C4 C2×D4×D5 C22×D5 C22.D4 D10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C2 # reps 1 3 2 1 2 2 1 1 1 1 1 4 2 8 6 4 2 2 4 8

Matrix representation of D5×C22.D4 in GL6(𝔽41)

 34 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 34 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 9 0 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 32 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 2 32 0 0 0 0 0 0 0 40 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 23 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,9,1,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,2,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,23,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

D5×C22.D4 in GAP, Magma, Sage, TeX

D_5\times C_2^2.D_4
% in TeX

G:=Group("D5xC2^2.D4");
// GroupNames label

G:=SmallGroup(320,1324);
// by ID

G=gap.SmallGroup(320,1324);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,346,297,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=d*e^-1>;
// generators/relations

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